In the EEZ-GCD algorithm [1]}, a factorization of the GCD of the leading coefficient is computed. However, this step may lead to high complexity, because the number of terms of the factors may be very large.
Our algorithm predetermines the leading coefficient of the GCD by isolating the maximum term instead of factorization, which was similar to the method introduced by Cuyt and Lee [2]} and had controllable complexity.
Furthermore, inspired by the work of Klivans and Spielman [3]},
we introduce a new variable substitution such that the leading coefficient of the substituted GCD is a monomial, which greatly decreases the computation cost.
Compared to the algorithm in [4]},
our algorithm also uses Ben-Or/Tiwari algorithm to interpolate the coefficients of GCD,
but our algorithm was based on a new diversification method.
Also, we do not use the Kronecker substitution and use only primes with small size.
These techniques allow us to give an exact bit complexity, while in their algorithm the smooth prime has size \(O(D^n)\)  in theory.
Compared to the algorithm in [5]},
our algorithm uses the method of isolating maximum term instead of the shifted homogenization introduced in [2]}
and our algorithm works for any finite field even if the degree is large.
